Question

Suppose a parallelogram is drawn on a coordinate plane. Using verticies with coordinates (2,1), (2,3), (4,4), and (4,2), compute the perimeter. Round your answer to the nearest tenth. (1 point) The perimeter is approximately units.

Answer 1

To find the perimeter of the parallelogram with the given vertices \((2,1)\), \((2,3)\), \((4,4)\), and \((4,2)\), we first need to calculate the lengths of the sides.

### Step 1: Identify the pairs of vertices:

We can pair the vertices as follows:
- Vertices A(2, 1) and B(2, 3)
- Vertices C(4, 4) and D(4, 2)

### Step 2: Calculate the lengths of the sides.

The lengths of the sides can be calculated using the distance formula for points \((x_1, y_1)\) and \((x_2, y_2)\):

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

#### Length of AB:
\[
AB = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 2^2} = \sqrt{4} = 2
\]

#### Length of BC:
\[
BC = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]

The lengths of the sides of a parallelogram opposite sides are equal. So:
- \(AB\) and \(DC\) are both of length \(2\).
- \(BC\) and \(AD\) are both of length \(\sqrt{5}\).

### Step 3: Calculate the perimeter.

The perimeter \(P\) of a parallelogram is given by:
\[
P = 2(AB + BC) = 2(2 + \sqrt{5})
\]

Calculating \(\sqrt{5}\):
\[
\sqrt{5} \approx 2.236
\]

Now substituting this into the perimeter:
\[
P \approx 2(2 + 2.236) = 2(4.236) \approx 8.472
\]

### Step 4: Round to the nearest tenth.

Rounding \(8.472\) gives:
\[
P \approx 8.5
\]

Thus, the perimeter of the parallelogram is approximately **8.5 units**.